Question

Use a conditional trigonometric equations calculator.

a) Solve for x in cos(x) = sin(2x)
b) Solve for θ in tan(θ) = 1
c) Find the values of x satisfying sin(x) = cos^2(x)
d) Solve for φ in sec(φ) = csc(2φ)

Answer 1

The question requires solving various trigonometric equations using identities and algebraic manipulations. Solutions involve finding specific angle measures where the equations hold true, using radians for angle measurement.

Step-by-step explanation:

The question involves solving trigonometric equations, a topic in high school mathematics. Let's solve each equation step by step:

1. cos(x) = sin(2x): To solve this, we use trigonometric identities. We know that sin(2x) = 2sin(x)cos(x); therefore, our equation becomes cos(x) = 2sin(x)cos(x). Dividing both sides by cos(x) (assuming cos(x) ≠ 0), we get 1 = 2sin(x). Thus, sin(x) = 1/2, which has solutions at x = π/6, 5π/6 (plus any multiple of 2π for general solutions).
2. tan(θ) = 1: This equation is true when θ is π/4, plus any multiple of π due to the periodicity of the tangent function.
3. sin(x) = cos^2(x): Using the Pythagorean identity, we can replace cos^2(x) with 1 - sin^2(x). This gives us the quadratic sin^2(x) + sin(x) - 1 = 0, which we can solve using the quadratic formula.
4. sec(ϕ) = csc(2ϕ): This involves an identity that sec(ϕ) = 1/cos(ϕ) and csc(2ϕ) = 1/sin(2ϕ). We can use the double-angle formula for sine here as well and solve for ϕ.

Remember to use units of radians for angles when calculating trigonometric equations, and always check your answer to see if it is reasonable.